- Email: [email protected]
Second order approximation of wave-structure couplings
Ocean
Engng,
Vol. 22. No.
I, pp. 49-64, 1995 Elsevier Science Ltd Printed in Great Britain 0029-X018/95 $7.00 + CC
Pergamon
SECOND ORDER APPROXIMATION WAVE-STRUCTURE COUPLINGS E. T. Huang*
and Yuh-Lin
*Naval Civil Engineering Laboratory, tconsultant, 10958 Barman Ave.,
OF
Hwangt
Port Hueneme, California, U.S.A.; Culver City, CA 90230, U.S.A.
Abstract-This study explores the significance of second order wave forces on a box-shaped pontoon in shallow water and tests the feasibility of the perturbation approach for free surface problems involving shallow draft structures. A simulation model developed for these purposes is described that solves the reduced linear boundary value problems with a versatile finite element procedure. This procedure, which achieves a fair compromise between computation costs and physical details, is particularly suitable for practical applications. A simple hydraulic model test was conducted to observe the wave forces imposed on a rectangle box by a steep wave in shallow water. The second order contribution to the total wave force was found to increase with the Ursell number, HLZlh3, in a manner resembling that of the second order waves to the exciting waves. The test results favorably support the perturbation approach for the analysis of nonlinear wave forces on shallow draft pontoons. However, more sophisticated model tests are required for a full justification. Both theoretical and experimental results show profound second order forces that are sufficient to impact the design of pontoon facilities.
INTRODUCTION
Linear assumptions underlying first order theories of free surface flows suppress certain nonlinear hydrodynamic details important to the design of ocean structures. These details can only be addressed with the retention of the nonlinear terms of the equation of motion and a proper specification of free surface conditions. Inclusion of these nonlinear effects in the diffraction waves would result in additional wave forces acting at double the frequency of the incident waves. These forces, despite their small magnitude in comparison with first order forces, are of major concern to the structural design if they act at a frequency near the natural frequency of the structure motions or where restoring or damping forces are small. Significant progress has been made recently in the development of fully nonlinear diffraction theories through the efforts of Maskew (1991), Korsmeyer et al. (1992), Cao et al. (1992) and Lin and Yue (1993), among others. These advanced theories are capable of analyzing large amplitude responses of floating bodies to severe seaways. However, these theories involve intensive computations that can only be executed on supercomputers and hence are expected to remain of more academic interest in the foreseeable future. A simplified method that could be run on desktop workstations would be more practical for a general analysis of a problem that involves a large number of physical parameters. Previous attempts with approximated methods can be found in works by Issacson (1977), Chakrabarti (1978), Molin (1979), Hunt and Baddour (1981), Chen and Hudspeth (1982), Rahman (1983) and Ogilvie (1983). The results are, however, controversial due to difficulties in the proper treatment of the 49
50
E. T. Huang
and Yuh-Lin
Hwang
second order free surface boundary condition and the radiation condition for the second order diffraction waves. Kim and Yue (1989) derived a more complete second order diffraction solution for an axisymmetric body in constant water depth using a perturbation expansion technique. The solution gives an explicit second order flow potential in terms of wave-source Green functions. Wave forces were calculated to the second order. The study presented here follows a similar approach in searching for a costeffective means to address a system of coupled floating pontoons over a sloping sea bottom. The associated boundary value problems were solved by an efficient finite element algorithm. PROBLEM
FORMULATION
A mathematical idealization of hydrodynamic couplings between waves and a group of large floating or fixed structures in water with a free surface is defined in Fig. 1. The water body may be open or partially sheltered, and the water depth may vary. Relevant waves, including incident, scattering, and radiation, are described in a way similar to Stokes’ second order theory. A Cartesian coordinate system, OXYZ, fixed to the Earth, as shown in Fig. 2, is employed as an inertia reference, where the X-Y plane lies on the mean water surface and the Z coordinate is measured positive upward from the still water. The responses of a floating body are described in reference to a body coordinate system, Gxyz, attached to the body with its origin G located at the center of gravity of the body. The x-y plane is parallel to the static waterplane, and the z coordinate is directed vertically upward. For the purpose of computing wave forces and moments exerted on the body, a third system of coordinate axes, G.i?ji, is
Fig. 1. Definition
sketch
of a compound
structure
and the ambient
fluid
Wave-structure
51
couplings
Body Coordinate System L Y
Coordinate
Fig. 2. Coordinate
systems.
defined where its origin is at the center of gravity of the body and is always parallel to the fixed coordinate system OXYZ. With the assumption that the fluid is inviscid and incompressible and that the flow is irrotational, the relevant waves can be addressed using potential theory. Further assuming weak nonlinear waves, the total velocity potential @ can be expanded in terms of the wave slope parameter I = kA 4 1 as follows: @ = &Q(l) + &2Q(2) + O(E3)
(1)
where Q(‘) and @(*) are first and second order velocity potentials. The incident wave number, k, is defined by the dispersion relationship, o* = gk tanh (kh), with g and w denoting the gravitational acceleration and the base frequency of incident waves. The associated free surface profile -rl is decomposed likewise: n = &n(l) + &2+)
+ O(E3) .
(2)
This expansion decomposes an otherwise nonlinear boundary value problem to a series of linear boundary value problems. For practical purposes, the series will be truncated at the second order. The first order problem is classical, and a variety of numerical solutions are available. The present study focuses on the second order problem. As usual, the second order velocity potentials comply with the Laplace equation V* Q(*) = 0 in the fluid domain and, in the meantime, satisfy the following boundary conditions: at the seabed:
@D(2) -=O;
(3)
an
at body surfaces: for scattered
and incident
waves:
a@(*) an = -(x(l)
. V)V@(r) . n ;
(4)
52
E. T. Huang
for radiation at the free surface:
and Yuh-Lin
Hwang
waves: @,I:’ + go,, (2) = -2V@(‘,
. V@,j’,
Radiation condition: outgoing waves from the floating bodies. In the above, x(l) is the first order displacement of the wetted surface with n being the unit normal vector pointing into water body. al&z is the normal derivative. The second order potentials are tied to the first order potentials in terms of quadratic products of the latter. Hence, @c2) is completely defined once @cl) is given. The first order velocity potential is conventionally decomposed in three components as follows: @(‘) = @$I’ + a.$*, + @&!I
(7)
where the subscripts I, S and B indicate the incident, scattering and radiation waves, respectively. Substituting these components into Equation (6), the second order free surface boundary condition can be reorganized in the following form: @$:I + g@&?) = Q(x,y,O)
eP2iti’r
(8)
The right-hand side of Equation (8) appears like a forcing function oscillating at a frequency twice the linear waves. This function was interpreted by Wehausen and Laitone (1960) as a nonuniform pressure field imposed on the free water surface. The pressure field leads to additional cylindrical standing and outwardly going progressive waves at the double frequency. The quadratic forcing function, Q(x,y,O), consists of nine components according to their sources as follows:
Q(KY,O) = Q,, + Qss + QBB + (Q1.s + es,> + (Qm + QRI) + (Qss + QBS) where QIl is the plane wave forcing function order waves acting at double frequency 20; first order scattered and radiation waves, cross-couplings among the first order waves. the following:
(9)
which leads to the ordinary Stokes’ second Qss and QBB represent self-couplings of respectively; and Qls-QBs represent the Details of these components are given in
Wave-structure
53
couplings
where the subscriptions of spatial coordinates indicate partial differential. Since Equation (8) is nonhomogeneous, the second order potential consists solution @(2)p as: homogeneous solution @(*jH and the particular @,(2) = @,(W + @WP . Each component surface boundary
(11)
satisfies the corresponding homogeneous or nonhomogeneous condition of Equation (8). The homogeneous solution consists
@(Wf = @*’ + @g’ ; and the particular W)P As a result,
solution
order
of: (13) potential
becomes:
= (I$’ + CDL*’+ @F’ + @$22
The component (I$) is the second order (15) for the case of constant water depth. cow2
free of: (12)
= @iF’ + @$?A .
the total second W)
consists
of the
k(d
(14)
Stokes’
+
sinh4( k d)
211 ’
wave,
which
is given
by Equation
(15)
The @& component results from the entire quadratic forcing functions, except for Q,,. It represents the coupling effects among the first order potentials. @$*I is the scattering wave component corresponding to @SF), and C$) is the radiation waves component due to double frequency vibrations of the floating body. Both satisfy the homogeneous free surface boundary condition. Their far-field behavior is given by: &-r
-!!eW
+ 0 r-4 , r + 1 i 1
(16)
where k2 is the double frequency wave number satisfying 4w2 = gk2 tanh (k,h). This implies that these components can be solved by the same procedure for the first order problems.
54
E. T. Huang FINITE
and Yuh-Lin
ELEMENT
Hwang
SOLUTION
According to the calculus of variations [see Hildebrand (1965), for example], the solution to a boundary value problem is the potential which minimizes a certain functional, in terms of the governing equation of the problem. For (P@), the functional ZP can be expressed as: 11(2, = iii
o
; (V P’Y
do + i/S +s +,~ f(V) f
B
dS
(17)
K
where R is the fluid domain, and S,, SB and SR are the free surface boundary, body boundary and radiation boundary, respectively. A radiation boundary condition, 12, is imposed at moderate distances from the structures of interest to define a finite fluid domain. The water body within this domain is divided into a three-dimensional (3D) region R,, and a two-dimensional (2D) region 0,. The 3D region includes areas near structures where the seafloor may change substantially, and the 2D region includes the remaining areas where the seafloor is reasonably flat. In 3D regions, waves are addressed in detail in 3D space to account for the effects due to irregular geometries of structures and the seafloor. In 2D regions, waves are approximated by plane wave theories in terms of free surface parameters following the mild slope concept of Berkhoff (1974) for better computation efficiency. Berkhoff’s mild slope equation has been extended to the second order to be compatible with the three-dimensional descriptions; the details will be addressed in a forthcoming paper by the same authors. Boundaries for these two regions are illustrated in Fig. 1. These boundaries are the radiation boundary, 12, and the border separating the two regions, I,. Waves transmitted beyond the radiation boundary are accounted for in a collective form in terms of parameters on the radiation boundary. Incident waves are specified on the wave input boundary Ii, which coincides with the radiation boundary for convenience, when wind wave excitations are considered. Other boundaries involved are the sea bottom, Ib, the wetted surfaces of floating structures, I,, and fixed structures, I,. Subdivisions of the water body and the associated boundaries are illustrated in Fig. 3. The functionals in each domain are given in terms of the corresponding governing equations and boundary conditions. For the three-dimensional region:
Water Surface
Elements
_
Fig. 3. Finite
element
mesh
Wave-structure
couplings
55
(18)
where subscript j represents the II, CC, S and B components. The last two integrals exist for radiation waves (B components) only, in which subscript i denotes six modes of body motion and subscript k is the identification number of the floating bodies. For instance, $1;) is the radiation waves due to motion in the ith mode by the kth body. skrn is the Kronecker delta. x is the first order displacement vector at the wetted body surface. n is the outward normal to the surface in body coordinates. The quantity Q in the third term is the quadratic forcing functions. For the two-dimensional region:
+ (radiation
condition)
, j = II, CC, S and B .
The function F”, which is a function resulting boundary condition [Equation (6)] for the second in the following: sinh 2 k2h
F”=
2k2H
from the nonlinear free surface order Berkhoff equation, is given
cf*)qw32~ ’
(20)
1 2
k2 H sinh(k2h)
- Fcosh
(k,h)
where
C(2) I
+w
W2 ,atz=O LL - -p t g = iI [ and H is the mean water depth of the two-dimensional domain. A boundary resembles the one derived by Bando et al., (1982) given in the following F
=
_i2$
-(vp))”
+ ~
jp
2
(21) integral
(22) is added to the functional given as:
for the two-dimensional
domain.
The constants
IY.and B are
E. T. Huang and Yuh-Lin Hwang
5h
a = [3/(4R2) p = 1/(2/R where region
- 2k$ + 3ik,lR]l(2/R
+ 2ik2)
+ 2ik2)
R is the radius of the radiation boundary, is coupled with the two-dimensional region
4j2’(qv)
r2. On J?,, the three-dimensional by the following relationship:
cash k2(h + z) ~~~, j=II,CC,SandB.
= 43hY)
(23)
sinh4 The fluid domain and boundaries are discretized in elements of finite size as shown in terms of in Fig. 3. The potential 4(2)’ within each element can be approximated unknown parameters {4i2)} at the nodes of a specific element and interpolation functions [N] as follows:
4(3 = [N] {4$“} .
(24)
This approximation digitizes the above functionals for best computation efficiency. The condition that the solution is the set of functions that minimizes the associated functionals further converts the integral differential equations to a standard linear algebraic equation. The result can be shown in a compact matrix equation as follows:
[kl {4$“1 - W = 0.
(25)
The element stiffness matrix [k] and the element nodal force vector {f} in Equation (25) for three- and two-dimensional elements are given in the following. For three-dimensional elements:
[kl =
j/l
(VN)‘(VN) 12
if) = - \/ + for two-dimensional
[kl = j/
N”(v,,),dS Sk
N’fdS
dx dy dz -
sf4;’
NTNdS
il . Sk,,, +
NT[(~(‘)V)V
;
4;” . n]dS . Sk,,,
(27)
elements: { (VN)TP(VN)
- 4 k2F*NTN}
dx dy
(28)
(29)
Wave-structure
57
couplings
In the above equations, integrations are taken over a single element only. Equations for elements can be assembled to form a set of global equations pertaining to the entire fluid region. These may be written in matrix form as:
[~I{#“~ = {W .
(30)
The global stiffness matrix [K] is symmetric and banded. The matrix solved by a proven procedure based on the frontal method developed SECOND
ORDER
WAVE
equation may be by Irons (1970).
FORCES
The second order wave forces are computed following a procedure proposed by Pinkster and Oortermersen (1977). The total pressure distribution on the wetted surfaces can be determined by substituting the second order velocity potentials in the Bernoulli’s equation. The pressure, accurate to the second order, is given by P’P
(“1 + ep (1) + &*p(*) + 0 (e3) .
(31)
These components are associated, respectively, second order pressures as given by the following P P
to the hydrostatic, expressions:
first order
(0) = - pgzco’ (1) = - pgz(”
and
(32) - p@yl
(33)
p(*) zz - 1 JV@(‘,]2 - o@‘jZ) - p(xU’ 2p
. V@,jl)) .
(34)
The total fluid force imposed on the body in the coordinate system G,i?ji is obtained by integrating the pressure over instantaneous wetted surface S as follows: FE-
The surface S is composed of the constant wetted surface up to the still water line So and the surface between the still water line and the wave profile along the vessel s, as described in Fig. 2. N is the instantaneous normal vector to the surface element dS relative to system G.i?ji. It is further assumed that the normal vector may be expanded in perturbation components as follows: N = N(O) + EN(‘) + O(e’) Carrying
the integration
.
in Equation
(36) (35) to the second
F = F(O) + c F(l) + &*F(*) + 0(c3) The second
order
contribution
I?(*) = -
@(‘)N(l) ii so
order,
one obtains:
.
(37)
to the total forces is: + pc2)n) dS -
p(l)ndS . ii s
The first term is induced by the first order waves, which are related motion in accordance to Newton’s second law of motion by:
(38) to the first order
58
E. T. Huang and Yuh-Lin Hwang
-
$1)
xg’
NC’) dS = f)(1) x [M]
(39)
where [M] is the mass of the vessel in air, and @) is the first order acceleration of the center of gravity of the body. The second term represents the contributions by the second order potentials, and the third term accounts for the force due to water surface variation from the mean elevation. The last two terms are evaluated by numerical integration. The result of the total second order force becomes:
I
F(2) = _
wL ; p g(i$‘))“n dl + 9(i) x [M] Xg’
-
The total hydrodynamic to the coordinate system M(2) = _ The second
order
1
- ; p ]V W’)]2 - p @s2) - p(X(‘) . V@,j’)) n dS moment about the center of gravitv of the vessel G.Qji is given by Equation (41): d
p(X x N) dS ii s
wave moment
M(2) = _ i WL _
(40) relative
(41)
is derived
similarly:
; P lttCL’J2 ( x x n) dl + f!)(l) x [I] tic’)
-
;
p IV @Cl)12
-
p cg”
EXPERIMENTAL
-
1
p(X(‘) . V @$I)) (x x n) dS
(42)
STUDY
A hydraulic mode1 was designed to measure the coupling forces on a fixed box by nonlinear waves in shallow water. The experiment was conducted in a two-dimensional wave tank at the Civil Engineering Laboratory of Texas A&M University. Figure 4 illustrates the genera1 layout of the experiment. The box was mounted to a rigid aluminum frame with three legs as shown in Fig. 4. The box was further ballasted to a neutrally buoyant condition to simulate a free-floating barge. Load cells were placed at the end of each of the three legs to measure the horizontal and vertical forces. The total forces and moments were deduced from these measurements. The experimental waves were generated by a computer-controlled hinged-flap wavemaker. The water pumped by the flap traveled down the tank channel and evolved into a steep shallow water wave before it reached the box. Two resistance-type wave gauges were used to monitor the wave profiles. Data were recorded using a Hewlett Packard HP-3852A data acquisition/control unit with an HP-330 workstation. Details of the test setup and procedures are described in an article by Hook et al. (1992). It was anticipated that the close proximity of side walls of the narrow wave tank would have a significant influence on the wave activity around the test model. The wave forces imposed on the test mode1 would be different from those experienced in
Wave-structure
59
couplings
loadcells SEE ( c ) FOR DEML 0 waw gauge
<
wave gauge
0
(a) top view
t--
LoAn CELLS
WAVE
-k
ABSORBER \
8.4 cm 10.3 cm I
I-
75 cm
47 cm
L
-~
/
11.9 cm
37M I(
(b) side view
29lcm
_
I
g33cm 762cm
i-
Fig. 4. General
layout
of the hydraulic
I
model
an otherwise open water. However, this test was intended to validate the numerical procedure. The side walls of the wave tank were simulated in the numerical procedure to give a fair comparison between the physical and numerical models. Twenty-four sets of time history of the wave profiles and the horizontal and vertical forces were recorded. A sample set of these measurements is shown in Fig. 5. The incident wave profile closely resembles that of theoretical Cnoidal waves in both wave height and length. However, it is noted that the experimental wave is asymmetric in contrast to the asymmetric mathematical form; this discrepancy grows as the wave
E. T. Huang
-12-1 21
and Yuh-Lin
Hwang
I 21.5
22
22.5
23
23.5
24
24.5
lime (SK)
-2x
: 21
Fig. 5. Profiles
I
I
21.5
22
of sample
incident
, 225 Tiie
ti (SCC)
wave height,
horizontal
23.5
24
force and vertical
24.5
force.
height increases. This is caused by the wave generation mechanism used in the test. The force time histories are similar in shape to those of the wave profile. The time histories of all measurements are decomposed by Fast Fourier transform up to the third harmonic as follows: X(f) = X,, +
5 + cos (not ,z-
I
+ e,)
(43)
Wave-structure
61
couplings
where X0 and X, are the mean value and the range of the nth harmonics, while o and e, are the base frequency and the associated phase. The wave height and the ranges of the forces in the horizontal and vertical directions are denoted by H, X and Z, respectively, in Fig. 6. The first and second harmonics of the forces and incident wave height are nondimensionalized by their respective resultants. These results are plotted vs the Ursell number, U, = HL2/h3, which is a measure of the wave steepness and the relative water depth; h denotes water depth. 1.0 -
0
0.9 ‘-
‘(y
0’
,
n’
u
-
0.8 .’ 0.7 ‘. 0.6 ..
E .S ?=
?T
0.4,.
d
0.5
0.3 .-
1
> nmry
-
Sz
n=l
I n =2
0
0.2 *’
0.1.. 0.0
0
20
10
30
Ursell Number
1.0 1
8 $
0.9 ”‘. . 0.8
:
0.7 ..
z 5
0.6 ‘.
.p= iz
‘0
40
I
A
0.0
80
I
II=
0
0
cl
B
1
1
0
0
cm
0
0
.-
0.2 t 0.1
I 70
I
0
8:’ 0.3 ‘-w
t 60
HL2/ h3
“0 8
0
50
n =2
Ll I
0
I
10
20 Numbe?HL2/
40
Ursell I
Ill-i
I
t
50
60
70
80
I
I
I
I
h3 I
1
1
0.8.8
0 n=l
O
0
0
0
03 . . 0.2 .0.1.. 0.0
n=2 w
0
EF
‘00
T
20
10
Ursell Number Fig. 6. Fourier
decomposition
of incident
I 30
0’ 40
0‘ 1 50
, 60
q 70
80
HL2/ h3
waves and wave-induced
force as a function
of the Ursell number.
62
E. T. Huang
and Yuh-Lin
Hwang
The first harmonic ratios exhibit a linear trend, while the second harmonic ratios present some nonlinearity. HI/H and X,/X decrease linearly as the Ursell number increases, while Zi/Z remains constant. H,IH and X,/X increase initially then sustain a constant value, while Zz/Z decreases slightly throughout the range of Ursell numbers. NUMERICAL
RESULTS
The finite element procedure derived in the theoretical sections was executed through a computer program, NAUTILUS, which is coded in standard Fortran 77 language. This program was used to simulate the hydraulic model test described in the experimental section. Since NAUTILUS does not include a wave generation mechanism to simulate the wavemaker, the wave tank was approximated with a channel of infinite length with a steady second order Stokes’ wave approaching from one end of the channel. The water body within three times of the barge length from the center of the barge was simulated with three-dimensional element meshes, while the rest was approximated with two-dimensional element meshes. The water body under consideration extended to a distance of 10 barge lengths on each side. Figure 7 illustrates the discretization of the model test setup. Wave parameters used in the model test were entered into the numerical model for comparison with the forces observed from the model test. The results of the wave forces predicted by the numerical model (in black symbols) are presented along with the hydraulic model measurements (in white symbols) in Fig. 6. The fair agreement between the theoretical and the empirical results supports the approach taken.
Fig. 7. Finite
element
mesh of the hydraulic
model
Wave-structure
couplings
63
SUMMARY
A second order simulation model for three-dimensional wave-structural couplings in moderate seas has been formulated based on a perturbation approach. This approach allows the nonlinear problem to be approximated by linear components in terms of parameters at the mean water surface. The components were solved effectively with a finite element procedure. This procedure, which achieves a fair compromise between computation efficiency and the physical reality by using a combination of two- and three-dimensional finite elements, is particularly suitable for practical applications. The wave-induced forces on a rectangular box held fixed at the water surface were studied both numerically and experimentally to examine the significance of the second order forces. The experimental results show favorable support of the perturbation approach for nonlinear free surface problems involving shallow draft pontoon structures. The second order contribution to the total wave forces was found to increase with the Ursell number, HL2/h3, in a manner closely resembling that of second order waves to the exciting waves. Hydraulic model measurements indicate the second order lateral forces in shallow water can be as large as 30% of the total lateral force. Their magnitudes are sufficient to cause severe adverse consequences to a structure. More importantly, these forces act at a frequency away from the design wave frequency, and may coincide with the resonance frequency of a structure. Ocean facilities for use in high sea states must be designed accordingly. Acknowledgement-This project the Office of Naval Research.
was conducted
under
NCEL
IR/IED
Program,
ROOO-NO-210, sponsored
by
REFERENCES Bando, K., Bettess, P. and Emson, C. 1982. The effectiveness of dampers for the analysis of exterior scalar wave diffraction by cylinders and ellipsoids. Report C/R/430/82. Berkhoff, J.C.W. 1974. Linear wave propagation problems and the finite element method. In Finite Elements in Fluids, Gallegher, R.H. et al. (eds), Vol. 1, pp. 251-280. John Wiley & Sons, London. Cao, Y., Lee, T. and Beck, R. F. 1992. Computation of nonlinear waves generated by floating bodies. 7th International Workshop on Water Waves and Floating Bodies, Val de Riuil, France. Chakrabarti, S.K. 1978. Comments on second-order wave effects on a large diameter vertical cylinder. J. Ship Res. 22, 266-268. Chen, M.C. and Hudspeth, R.T. 1982. Nonlinear diffraction by eigenfunction expansions. J. WatWay, Port, Coastal Ocean Div., ASCE, 108, 306-325. Hildebrand, F.B. 1965. Method of Applied Mathematics (2nd Edn). Prentice-Hall, New Jersey. Hook, G.S., Kim, C.H. and Huang, E.T. 1992. Wave excitating forces on a platform fixed in nonlinear shallow water waves. In Proceedings of the Internarional Conference, Civil Engineering in the Oceans V, College Station, Texas, 2-5 November, pp. 311-325. Hunt, J.N. and Baddour, R.E. 1981. The diffraction of nonlinear progressive waves by a vertical cylinder. Q. J. Mech. Appl. Math. 34, 69-87. Irons, B.B. 1970. A frontal solution program for finite element analysis. Int. J. Numerical Math. Engng 2, 5-32. Kim, M.H. and Yue, D.K.P. 1989. The complete second-order diffraction solution for an axisymmetric body, Part 1. Monochromatic incident waves. J. Fluid Mech. 200, 235-264. Korsmeyer, F.T., Ma, C., Xu, H. and Yue, D.K.P. 1992. The fully nonlinear diffraction waves by a surface piercing strut. Proceedings of the 19th Symposium on Naval Hydrodynamics, Seoul, Korea. Issacson, M.Q. 1977. Nonlinear wave forces on large offshore structures. J. WarWay, Port, Coastal Ocean Div., ASCE 103, 166-170. Lin, W.M. and Yue, D.K.P. 1993. Time-domain analysis for floating bodies in mild-slope waves of largeamplitude. 8th Inlernarional Workshop on Wave Waves and Floating Bodies, St Johns, Newfoundland. Maskew, B. 1991. A nonlinear numerical method for transient wave/hull problems on arbitrary vessels. Trans. Sot. nav. Archit. Mar. Engrs, 99.
64
E. T. Huang
and Yuh-Lin
Hwang
Molin, B. 1979. Second-order diffraction loads upon three-dimensional bodies. Appl. Ocean Res. 1, 197-202. Ogilvie, T.F. 1983. Second-order hydrodynamic effects on ocean platforms. International Workshop on Ship and Platform Motion, Berkeley, pp. 205-265. Pinkster, J.k. and Van Oortermersen, G. 1977. Computation of the first and second order wave forces on oscillating bodies in regular waves. 2nd International Conference on Numerical Shir, 1 Hvdrodvnamics. _ L University of California, Berkeley. Rahman, M. 1983. Wave diffraction by large offshore structures; an exact second-order theory. Appl. Ocean Res. 6, 90-100. Wehausen, J.V. and Laitone, E.V. 1960. Surface wave. In Handbuch des Physik, Vol. 9, Springer, Berlin. pp. 446-778.
Engng,
Vol. 22. No.
I, pp. 49-64, 1995 Elsevier Science Ltd Printed in Great Britain 0029-X018/95 $7.00 + CC
Pergamon
SECOND ORDER APPROXIMATION WAVE-STRUCTURE COUPLINGS E. T. Huang*
and Yuh-Lin
*Naval Civil Engineering Laboratory, tconsultant, 10958 Barman Ave.,
OF
Hwangt
Port Hueneme, California, U.S.A.; Culver City, CA 90230, U.S.A.
Abstract-This study explores the significance of second order wave forces on a box-shaped pontoon in shallow water and tests the feasibility of the perturbation approach for free surface problems involving shallow draft structures. A simulation model developed for these purposes is described that solves the reduced linear boundary value problems with a versatile finite element procedure. This procedure, which achieves a fair compromise between computation costs and physical details, is particularly suitable for practical applications. A simple hydraulic model test was conducted to observe the wave forces imposed on a rectangle box by a steep wave in shallow water. The second order contribution to the total wave force was found to increase with the Ursell number, HLZlh3, in a manner resembling that of the second order waves to the exciting waves. The test results favorably support the perturbation approach for the analysis of nonlinear wave forces on shallow draft pontoons. However, more sophisticated model tests are required for a full justification. Both theoretical and experimental results show profound second order forces that are sufficient to impact the design of pontoon facilities.
INTRODUCTION
Linear assumptions underlying first order theories of free surface flows suppress certain nonlinear hydrodynamic details important to the design of ocean structures. These details can only be addressed with the retention of the nonlinear terms of the equation of motion and a proper specification of free surface conditions. Inclusion of these nonlinear effects in the diffraction waves would result in additional wave forces acting at double the frequency of the incident waves. These forces, despite their small magnitude in comparison with first order forces, are of major concern to the structural design if they act at a frequency near the natural frequency of the structure motions or where restoring or damping forces are small. Significant progress has been made recently in the development of fully nonlinear diffraction theories through the efforts of Maskew (1991), Korsmeyer et al. (1992), Cao et al. (1992) and Lin and Yue (1993), among others. These advanced theories are capable of analyzing large amplitude responses of floating bodies to severe seaways. However, these theories involve intensive computations that can only be executed on supercomputers and hence are expected to remain of more academic interest in the foreseeable future. A simplified method that could be run on desktop workstations would be more practical for a general analysis of a problem that involves a large number of physical parameters. Previous attempts with approximated methods can be found in works by Issacson (1977), Chakrabarti (1978), Molin (1979), Hunt and Baddour (1981), Chen and Hudspeth (1982), Rahman (1983) and Ogilvie (1983). The results are, however, controversial due to difficulties in the proper treatment of the 49
50
E. T. Huang
and Yuh-Lin
Hwang
second order free surface boundary condition and the radiation condition for the second order diffraction waves. Kim and Yue (1989) derived a more complete second order diffraction solution for an axisymmetric body in constant water depth using a perturbation expansion technique. The solution gives an explicit second order flow potential in terms of wave-source Green functions. Wave forces were calculated to the second order. The study presented here follows a similar approach in searching for a costeffective means to address a system of coupled floating pontoons over a sloping sea bottom. The associated boundary value problems were solved by an efficient finite element algorithm. PROBLEM
FORMULATION
A mathematical idealization of hydrodynamic couplings between waves and a group of large floating or fixed structures in water with a free surface is defined in Fig. 1. The water body may be open or partially sheltered, and the water depth may vary. Relevant waves, including incident, scattering, and radiation, are described in a way similar to Stokes’ second order theory. A Cartesian coordinate system, OXYZ, fixed to the Earth, as shown in Fig. 2, is employed as an inertia reference, where the X-Y plane lies on the mean water surface and the Z coordinate is measured positive upward from the still water. The responses of a floating body are described in reference to a body coordinate system, Gxyz, attached to the body with its origin G located at the center of gravity of the body. The x-y plane is parallel to the static waterplane, and the z coordinate is directed vertically upward. For the purpose of computing wave forces and moments exerted on the body, a third system of coordinate axes, G.i?ji, is
Fig. 1. Definition
sketch
of a compound
structure
and the ambient
fluid
Wave-structure
51
couplings
Body Coordinate System L Y
Coordinate
Fig. 2. Coordinate
systems.
defined where its origin is at the center of gravity of the body and is always parallel to the fixed coordinate system OXYZ. With the assumption that the fluid is inviscid and incompressible and that the flow is irrotational, the relevant waves can be addressed using potential theory. Further assuming weak nonlinear waves, the total velocity potential @ can be expanded in terms of the wave slope parameter I = kA 4 1 as follows: @ = &Q(l) + &2Q(2) + O(E3)
(1)
where Q(‘) and @(*) are first and second order velocity potentials. The incident wave number, k, is defined by the dispersion relationship, o* = gk tanh (kh), with g and w denoting the gravitational acceleration and the base frequency of incident waves. The associated free surface profile -rl is decomposed likewise: n = &n(l) + &2+)
+ O(E3) .
(2)
This expansion decomposes an otherwise nonlinear boundary value problem to a series of linear boundary value problems. For practical purposes, the series will be truncated at the second order. The first order problem is classical, and a variety of numerical solutions are available. The present study focuses on the second order problem. As usual, the second order velocity potentials comply with the Laplace equation V* Q(*) = 0 in the fluid domain and, in the meantime, satisfy the following boundary conditions: at the seabed:
@D(2) -=O;
(3)
an
at body surfaces: for scattered
and incident
waves:
a@(*) an = -(x(l)
. V)V@(r) . n ;
(4)
52
E. T. Huang
for radiation at the free surface:
and Yuh-Lin
Hwang
waves: @,I:’ + go,, (2) = -2V@(‘,
. V@,j’,
Radiation condition: outgoing waves from the floating bodies. In the above, x(l) is the first order displacement of the wetted surface with n being the unit normal vector pointing into water body. al&z is the normal derivative. The second order potentials are tied to the first order potentials in terms of quadratic products of the latter. Hence, @c2) is completely defined once @cl) is given. The first order velocity potential is conventionally decomposed in three components as follows: @(‘) = @$I’ + a.$*, + @&!I
(7)
where the subscripts I, S and B indicate the incident, scattering and radiation waves, respectively. Substituting these components into Equation (6), the second order free surface boundary condition can be reorganized in the following form: @$:I + g@&?) = Q(x,y,O)
eP2iti’r
(8)
The right-hand side of Equation (8) appears like a forcing function oscillating at a frequency twice the linear waves. This function was interpreted by Wehausen and Laitone (1960) as a nonuniform pressure field imposed on the free water surface. The pressure field leads to additional cylindrical standing and outwardly going progressive waves at the double frequency. The quadratic forcing function, Q(x,y,O), consists of nine components according to their sources as follows:
Q(KY,O) = Q,, + Qss + QBB + (Q1.s + es,> + (Qm + QRI) + (Qss + QBS) where QIl is the plane wave forcing function order waves acting at double frequency 20; first order scattered and radiation waves, cross-couplings among the first order waves. the following:
(9)
which leads to the ordinary Stokes’ second Qss and QBB represent self-couplings of respectively; and Qls-QBs represent the Details of these components are given in
Wave-structure
53
couplings
where the subscriptions of spatial coordinates indicate partial differential. Since Equation (8) is nonhomogeneous, the second order potential consists solution @(2)p as: homogeneous solution @(*jH and the particular @,(2) = @,(W + @WP . Each component surface boundary
(11)
satisfies the corresponding homogeneous or nonhomogeneous condition of Equation (8). The homogeneous solution consists
@(Wf = @*’ + @g’ ; and the particular W)P As a result,
solution
order
of: (13) potential
becomes:
= (I$’ + CDL*’+ @F’ + @$22
The component (I$) is the second order (15) for the case of constant water depth. cow2
free of: (12)
= @iF’ + @$?A .
the total second W)
consists
of the
k(d
(14)
Stokes’
+
sinh4( k d)
211 ’
wave,
which
is given
by Equation
(15)
The @& component results from the entire quadratic forcing functions, except for Q,,. It represents the coupling effects among the first order potentials. @$*I is the scattering wave component corresponding to @SF), and C$) is the radiation waves component due to double frequency vibrations of the floating body. Both satisfy the homogeneous free surface boundary condition. Their far-field behavior is given by: &-r
-!!eW
+ 0 r-4 , r + 1 i 1
(16)
where k2 is the double frequency wave number satisfying 4w2 = gk2 tanh (k,h). This implies that these components can be solved by the same procedure for the first order problems.
54
E. T. Huang FINITE
and Yuh-Lin
ELEMENT
Hwang
SOLUTION
According to the calculus of variations [see Hildebrand (1965), for example], the solution to a boundary value problem is the potential which minimizes a certain functional, in terms of the governing equation of the problem. For (P@), the functional ZP can be expressed as: 11(2, = iii
o
; (V P’Y
do + i/S +s +,~ f(V) f
B
dS
(17)
K
where R is the fluid domain, and S,, SB and SR are the free surface boundary, body boundary and radiation boundary, respectively. A radiation boundary condition, 12, is imposed at moderate distances from the structures of interest to define a finite fluid domain. The water body within this domain is divided into a three-dimensional (3D) region R,, and a two-dimensional (2D) region 0,. The 3D region includes areas near structures where the seafloor may change substantially, and the 2D region includes the remaining areas where the seafloor is reasonably flat. In 3D regions, waves are addressed in detail in 3D space to account for the effects due to irregular geometries of structures and the seafloor. In 2D regions, waves are approximated by plane wave theories in terms of free surface parameters following the mild slope concept of Berkhoff (1974) for better computation efficiency. Berkhoff’s mild slope equation has been extended to the second order to be compatible with the three-dimensional descriptions; the details will be addressed in a forthcoming paper by the same authors. Boundaries for these two regions are illustrated in Fig. 1. These boundaries are the radiation boundary, 12, and the border separating the two regions, I,. Waves transmitted beyond the radiation boundary are accounted for in a collective form in terms of parameters on the radiation boundary. Incident waves are specified on the wave input boundary Ii, which coincides with the radiation boundary for convenience, when wind wave excitations are considered. Other boundaries involved are the sea bottom, Ib, the wetted surfaces of floating structures, I,, and fixed structures, I,. Subdivisions of the water body and the associated boundaries are illustrated in Fig. 3. The functionals in each domain are given in terms of the corresponding governing equations and boundary conditions. For the three-dimensional region:
Water Surface
Elements
_
Fig. 3. Finite
element
mesh
Wave-structure
couplings
55
(18)
where subscript j represents the II, CC, S and B components. The last two integrals exist for radiation waves (B components) only, in which subscript i denotes six modes of body motion and subscript k is the identification number of the floating bodies. For instance, $1;) is the radiation waves due to motion in the ith mode by the kth body. skrn is the Kronecker delta. x is the first order displacement vector at the wetted body surface. n is the outward normal to the surface in body coordinates. The quantity Q in the third term is the quadratic forcing functions. For the two-dimensional region:
+ (radiation
condition)
, j = II, CC, S and B .
The function F”, which is a function resulting boundary condition [Equation (6)] for the second in the following: sinh 2 k2h
F”=
2k2H
from the nonlinear free surface order Berkhoff equation, is given
cf*)qw32~ ’
(20)
1 2
k2 H sinh(k2h)
- Fcosh
(k,h)
where
C(2) I
+w
W2 ,atz=O LL - -p t g = iI [ and H is the mean water depth of the two-dimensional domain. A boundary resembles the one derived by Bando et al., (1982) given in the following F
=
_i2$
-(vp))”
+ ~
jp
2
(21) integral
(22) is added to the functional given as:
for the two-dimensional
domain.
The constants
IY.and B are
E. T. Huang and Yuh-Lin Hwang
5h
a = [3/(4R2) p = 1/(2/R where region
- 2k$ + 3ik,lR]l(2/R
+ 2ik2)
+ 2ik2)
R is the radius of the radiation boundary, is coupled with the two-dimensional region
4j2’(qv)
r2. On J?,, the three-dimensional by the following relationship:
cash k2(h + z) ~~~, j=II,CC,SandB.
= 43hY)
(23)
sinh4 The fluid domain and boundaries are discretized in elements of finite size as shown in terms of in Fig. 3. The potential 4(2)’ within each element can be approximated unknown parameters {4i2)} at the nodes of a specific element and interpolation functions [N] as follows:
4(3 = [N] {4$“} .
(24)
This approximation digitizes the above functionals for best computation efficiency. The condition that the solution is the set of functions that minimizes the associated functionals further converts the integral differential equations to a standard linear algebraic equation. The result can be shown in a compact matrix equation as follows:
[kl {4$“1 - W = 0.
(25)
The element stiffness matrix [k] and the element nodal force vector {f} in Equation (25) for three- and two-dimensional elements are given in the following. For three-dimensional elements:
[kl =
j/l
(VN)‘(VN) 12
if) = - \/ + for two-dimensional
[kl = j/
N”(v,,),dS Sk
N’fdS
dx dy dz -
sf4;’
NTNdS
il . Sk,,, +
NT[(~(‘)V)V
;
4;” . n]dS . Sk,,,
(27)
elements: { (VN)TP(VN)
- 4 k2F*NTN}
dx dy
(28)
(29)
Wave-structure
57
couplings
In the above equations, integrations are taken over a single element only. Equations for elements can be assembled to form a set of global equations pertaining to the entire fluid region. These may be written in matrix form as:
[~I{#“~ = {W .
(30)
The global stiffness matrix [K] is symmetric and banded. The matrix solved by a proven procedure based on the frontal method developed SECOND
ORDER
WAVE
equation may be by Irons (1970).
FORCES
The second order wave forces are computed following a procedure proposed by Pinkster and Oortermersen (1977). The total pressure distribution on the wetted surfaces can be determined by substituting the second order velocity potentials in the Bernoulli’s equation. The pressure, accurate to the second order, is given by P’P
(“1 + ep (1) + &*p(*) + 0 (e3) .
(31)
These components are associated, respectively, second order pressures as given by the following P P
to the hydrostatic, expressions:
first order
(0) = - pgzco’ (1) = - pgz(”
and
(32) - p@yl
(33)
p(*) zz - 1 JV@(‘,]2 - o@‘jZ) - p(xU’ 2p
. V@,jl)) .
(34)
The total fluid force imposed on the body in the coordinate system G,i?ji is obtained by integrating the pressure over instantaneous wetted surface S as follows: FE-
The surface S is composed of the constant wetted surface up to the still water line So and the surface between the still water line and the wave profile along the vessel s, as described in Fig. 2. N is the instantaneous normal vector to the surface element dS relative to system G.i?ji. It is further assumed that the normal vector may be expanded in perturbation components as follows: N = N(O) + EN(‘) + O(e’) Carrying
the integration
.
in Equation
(36) (35) to the second
F = F(O) + c F(l) + &*F(*) + 0(c3) The second
order
contribution
I?(*) = -
@(‘)N(l) ii so
order,
one obtains:
.
(37)
to the total forces is: + pc2)n) dS -
p(l)ndS . ii s
The first term is induced by the first order waves, which are related motion in accordance to Newton’s second law of motion by:
(38) to the first order
58
E. T. Huang and Yuh-Lin Hwang
-
$1)
xg’
NC’) dS = f)(1) x [M]
(39)
where [M] is the mass of the vessel in air, and @) is the first order acceleration of the center of gravity of the body. The second term represents the contributions by the second order potentials, and the third term accounts for the force due to water surface variation from the mean elevation. The last two terms are evaluated by numerical integration. The result of the total second order force becomes:
I
F(2) = _
wL ; p g(i$‘))“n dl + 9(i) x [M] Xg’
-
The total hydrodynamic to the coordinate system M(2) = _ The second
order
1
- ; p ]V W’)]2 - p @s2) - p(X(‘) . V@,j’)) n dS moment about the center of gravitv of the vessel G.Qji is given by Equation (41): d
p(X x N) dS ii s
wave moment
M(2) = _ i WL _
(40) relative
(41)
is derived
similarly:
; P lttCL’J2 ( x x n) dl + f!)(l) x [I] tic’)
-
;
p IV @Cl)12
-
p cg”
EXPERIMENTAL
-
1
p(X(‘) . V @$I)) (x x n) dS
(42)
STUDY
A hydraulic mode1 was designed to measure the coupling forces on a fixed box by nonlinear waves in shallow water. The experiment was conducted in a two-dimensional wave tank at the Civil Engineering Laboratory of Texas A&M University. Figure 4 illustrates the genera1 layout of the experiment. The box was mounted to a rigid aluminum frame with three legs as shown in Fig. 4. The box was further ballasted to a neutrally buoyant condition to simulate a free-floating barge. Load cells were placed at the end of each of the three legs to measure the horizontal and vertical forces. The total forces and moments were deduced from these measurements. The experimental waves were generated by a computer-controlled hinged-flap wavemaker. The water pumped by the flap traveled down the tank channel and evolved into a steep shallow water wave before it reached the box. Two resistance-type wave gauges were used to monitor the wave profiles. Data were recorded using a Hewlett Packard HP-3852A data acquisition/control unit with an HP-330 workstation. Details of the test setup and procedures are described in an article by Hook et al. (1992). It was anticipated that the close proximity of side walls of the narrow wave tank would have a significant influence on the wave activity around the test model. The wave forces imposed on the test mode1 would be different from those experienced in
Wave-structure
59
couplings
loadcells SEE ( c ) FOR DEML 0 waw gauge
<
wave gauge
0
(a) top view
t--
LoAn CELLS
WAVE
-k
ABSORBER \
8.4 cm 10.3 cm I
I-
75 cm
47 cm
L
-~
/
11.9 cm
37M I(
(b) side view
29lcm
_
I
g33cm 762cm
i-
Fig. 4. General
layout
of the hydraulic
I
model
an otherwise open water. However, this test was intended to validate the numerical procedure. The side walls of the wave tank were simulated in the numerical procedure to give a fair comparison between the physical and numerical models. Twenty-four sets of time history of the wave profiles and the horizontal and vertical forces were recorded. A sample set of these measurements is shown in Fig. 5. The incident wave profile closely resembles that of theoretical Cnoidal waves in both wave height and length. However, it is noted that the experimental wave is asymmetric in contrast to the asymmetric mathematical form; this discrepancy grows as the wave
E. T. Huang
-12-1 21
and Yuh-Lin
Hwang
I 21.5
22
22.5
23
23.5
24
24.5
lime (SK)
-2x
: 21
Fig. 5. Profiles
I
I
21.5
22
of sample
incident
, 225 Tiie
ti (SCC)
wave height,
horizontal
23.5
24
force and vertical
24.5
force.
height increases. This is caused by the wave generation mechanism used in the test. The force time histories are similar in shape to those of the wave profile. The time histories of all measurements are decomposed by Fast Fourier transform up to the third harmonic as follows: X(f) = X,, +
5 + cos (not ,z-
I
+ e,)
(43)
Wave-structure
61
couplings
where X0 and X, are the mean value and the range of the nth harmonics, while o and e, are the base frequency and the associated phase. The wave height and the ranges of the forces in the horizontal and vertical directions are denoted by H, X and Z, respectively, in Fig. 6. The first and second harmonics of the forces and incident wave height are nondimensionalized by their respective resultants. These results are plotted vs the Ursell number, U, = HL2/h3, which is a measure of the wave steepness and the relative water depth; h denotes water depth. 1.0 -
0
0.9 ‘-
‘(y
0’
,
n’
u
-
0.8 .’ 0.7 ‘. 0.6 ..
E .S ?=
?T
0.4,.
d
0.5
0.3 .-
1
> nmry
-
Sz
n=l
I n =2
0
0.2 *’
0.1.. 0.0
0
20
10
30
Ursell Number
1.0 1
8 $
0.9 ”‘. . 0.8
:
0.7 ..
z 5
0.6 ‘.
.p= iz
‘0
40
I
A
0.0
80
I
II=
0
0
cl
B
1
1
0
0
cm
0
0
.-
0.2 t 0.1
I 70
I
0
8:’ 0.3 ‘-w
t 60
HL2/ h3
“0 8
0
50
n =2
Ll I
0
I
10
20 Numbe?HL2/
40
Ursell I
Ill-i
I
t
50
60
70
80
I
I
I
I
h3 I
1
1
0.8.8
0 n=l
O
0
0
0
03 . . 0.2 .0.1.. 0.0
n=2 w
0
EF
‘00
T
20
10
Ursell Number Fig. 6. Fourier
decomposition
of incident
I 30
0’ 40
0‘ 1 50
, 60
q 70
80
HL2/ h3
waves and wave-induced
force as a function
of the Ursell number.
62
E. T. Huang
and Yuh-Lin
Hwang
The first harmonic ratios exhibit a linear trend, while the second harmonic ratios present some nonlinearity. HI/H and X,/X decrease linearly as the Ursell number increases, while Zi/Z remains constant. H,IH and X,/X increase initially then sustain a constant value, while Zz/Z decreases slightly throughout the range of Ursell numbers. NUMERICAL
RESULTS
The finite element procedure derived in the theoretical sections was executed through a computer program, NAUTILUS, which is coded in standard Fortran 77 language. This program was used to simulate the hydraulic model test described in the experimental section. Since NAUTILUS does not include a wave generation mechanism to simulate the wavemaker, the wave tank was approximated with a channel of infinite length with a steady second order Stokes’ wave approaching from one end of the channel. The water body within three times of the barge length from the center of the barge was simulated with three-dimensional element meshes, while the rest was approximated with two-dimensional element meshes. The water body under consideration extended to a distance of 10 barge lengths on each side. Figure 7 illustrates the discretization of the model test setup. Wave parameters used in the model test were entered into the numerical model for comparison with the forces observed from the model test. The results of the wave forces predicted by the numerical model (in black symbols) are presented along with the hydraulic model measurements (in white symbols) in Fig. 6. The fair agreement between the theoretical and the empirical results supports the approach taken.
Fig. 7. Finite
element
mesh of the hydraulic
model
Wave-structure
couplings
63
SUMMARY
A second order simulation model for three-dimensional wave-structural couplings in moderate seas has been formulated based on a perturbation approach. This approach allows the nonlinear problem to be approximated by linear components in terms of parameters at the mean water surface. The components were solved effectively with a finite element procedure. This procedure, which achieves a fair compromise between computation efficiency and the physical reality by using a combination of two- and three-dimensional finite elements, is particularly suitable for practical applications. The wave-induced forces on a rectangular box held fixed at the water surface were studied both numerically and experimentally to examine the significance of the second order forces. The experimental results show favorable support of the perturbation approach for nonlinear free surface problems involving shallow draft pontoon structures. The second order contribution to the total wave forces was found to increase with the Ursell number, HL2/h3, in a manner closely resembling that of second order waves to the exciting waves. Hydraulic model measurements indicate the second order lateral forces in shallow water can be as large as 30% of the total lateral force. Their magnitudes are sufficient to cause severe adverse consequences to a structure. More importantly, these forces act at a frequency away from the design wave frequency, and may coincide with the resonance frequency of a structure. Ocean facilities for use in high sea states must be designed accordingly. Acknowledgement-This project the Office of Naval Research.
was conducted
under
NCEL
IR/IED
Program,
ROOO-NO-210, sponsored
by
REFERENCES Bando, K., Bettess, P. and Emson, C. 1982. The effectiveness of dampers for the analysis of exterior scalar wave diffraction by cylinders and ellipsoids. Report C/R/430/82. Berkhoff, J.C.W. 1974. Linear wave propagation problems and the finite element method. In Finite Elements in Fluids, Gallegher, R.H. et al. (eds), Vol. 1, pp. 251-280. John Wiley & Sons, London. Cao, Y., Lee, T. and Beck, R. F. 1992. Computation of nonlinear waves generated by floating bodies. 7th International Workshop on Water Waves and Floating Bodies, Val de Riuil, France. Chakrabarti, S.K. 1978. Comments on second-order wave effects on a large diameter vertical cylinder. J. Ship Res. 22, 266-268. Chen, M.C. and Hudspeth, R.T. 1982. Nonlinear diffraction by eigenfunction expansions. J. WatWay, Port, Coastal Ocean Div., ASCE, 108, 306-325. Hildebrand, F.B. 1965. Method of Applied Mathematics (2nd Edn). Prentice-Hall, New Jersey. Hook, G.S., Kim, C.H. and Huang, E.T. 1992. Wave excitating forces on a platform fixed in nonlinear shallow water waves. In Proceedings of the Internarional Conference, Civil Engineering in the Oceans V, College Station, Texas, 2-5 November, pp. 311-325. Hunt, J.N. and Baddour, R.E. 1981. The diffraction of nonlinear progressive waves by a vertical cylinder. Q. J. Mech. Appl. Math. 34, 69-87. Irons, B.B. 1970. A frontal solution program for finite element analysis. Int. J. Numerical Math. Engng 2, 5-32. Kim, M.H. and Yue, D.K.P. 1989. The complete second-order diffraction solution for an axisymmetric body, Part 1. Monochromatic incident waves. J. Fluid Mech. 200, 235-264. Korsmeyer, F.T., Ma, C., Xu, H. and Yue, D.K.P. 1992. The fully nonlinear diffraction waves by a surface piercing strut. Proceedings of the 19th Symposium on Naval Hydrodynamics, Seoul, Korea. Issacson, M.Q. 1977. Nonlinear wave forces on large offshore structures. J. WarWay, Port, Coastal Ocean Div., ASCE 103, 166-170. Lin, W.M. and Yue, D.K.P. 1993. Time-domain analysis for floating bodies in mild-slope waves of largeamplitude. 8th Inlernarional Workshop on Wave Waves and Floating Bodies, St Johns, Newfoundland. Maskew, B. 1991. A nonlinear numerical method for transient wave/hull problems on arbitrary vessels. Trans. Sot. nav. Archit. Mar. Engrs, 99.
64
E. T. Huang
and Yuh-Lin
Hwang
Molin, B. 1979. Second-order diffraction loads upon three-dimensional bodies. Appl. Ocean Res. 1, 197-202. Ogilvie, T.F. 1983. Second-order hydrodynamic effects on ocean platforms. International Workshop on Ship and Platform Motion, Berkeley, pp. 205-265. Pinkster, J.k. and Van Oortermersen, G. 1977. Computation of the first and second order wave forces on oscillating bodies in regular waves. 2nd International Conference on Numerical Shir, 1 Hvdrodvnamics. _ L University of California, Berkeley. Rahman, M. 1983. Wave diffraction by large offshore structures; an exact second-order theory. Appl. Ocean Res. 6, 90-100. Wehausen, J.V. and Laitone, E.V. 1960. Surface wave. In Handbuch des Physik, Vol. 9, Springer, Berlin. pp. 446-778.